On the free energy in systems with separable interactions. III

In this paper we extend the calculation of the free energy in systems with separable interactions of the ferromagnetic type as given in paper I of this series to a more general class of systems characterized by a hamiltonian, in which the one-particle operator Σ_kT(k) is replaced by a more general operator T which may contain interactions as well. This extension enables us to simplify a great deal the derivation given in paper II of the lower bound to the free energy for systems with both ferromagnetic and antiferromagnetic interactions. We also discuss in some detail under which conditions the upper and the lower bounds can be equal. (The conditions include in particular the hamiltonian treated in paper II.)     

On-shell T-matrices in multiple scattering

The transition operator T for the scattering of a particle from N potentials V_j(x) can be expanded into a series featuring the transition operators t_j associated with the individual potentials. For V_j(x) both absolutely and square integrable in x, we show, using an analytic continuation argument, that if T is on-shell, i.e. in 〈k|T(k₀²±i0)|k′〉, |k| = |k′| = k₀, then each t_j is also on-shell.     

Convex-envelope formulation for separable many-particle interactions

The exact result for the free energy per particle in systems described by a hamiltonian of the type NP(V_N), where NV_N denotes a set of short-range operators, is reformulated in terms of a convex-envelope construction. A comparison is given with results obtained for classical systems with interactions of the so-called Kac-type.     

Systems with separable many-particle interactions. II

The treatment of a previous paper on systems with many-particle interactions is generalized to hamiltonians containing an analytic function of a number of short-range interaction operators V, which are normalized. An exact expression for the free energy per particle in the thermodynamic limit can be obtained from a trial hamiltonian which is linear in the operators V.     

One-dimensional hypervirial theorems in density functional theory

For a restricted class of operators hypervirial theorems are established involving wavefunctions ψ_i for N fermions which move independently in a common external one-dimensional potential V(x). Using this class it is possible to perform the summation over states ψ_i yielding exact relations between ground state density ? and kinetic energy density ?_k and potential V(x). It is checked to what extent the Thomas-Fermi expressions for ? and ?_k satisfy these relations.     

Direct application of the quasiparticle method to a three-nucleon system

The energy dependence of the expectation values of powers of the Lippmann-Schwinger operator,I _k (W)=〈γ|V(G ₀(W)·V) ^k |γ〉, is used to calculate the ground-state energy of the triton for a simple central two-nucleon interaction of Gaussian type. This method allows to calculate the exact energy eigenvalueE ₀ of a three-nucleon-system in principle with any desired precision. The calculations are performed untilk=3. In this approximation the valueE ₀ ⁽³=?13.25 MeV is obtained which is simultaneously an upper bound of the true groundstate energyE ₀ of the system.     

Temperature dependence of the radiation-stimulated conductivity of CsI crystals upon picosecond excitation by electron beams

The temperature dependence of the pulse conductivity for CsI crystals upon excitation with an electron beam (0.2 MeV, 50 ps, 400 A/cm²) at a time resolution of 150 ps is investigated. Under experimental conditions, the time of bimolecular recombination of electrons and holes (V _k centers) is directly measured in the temperature range 100–300 K. This made it possible to calculate the temperature dependence of the effective recombination cross section S(T)=7.9×10^?8 T² cm². The temperature dependence of the conductivity σ(T) is interpreted within the model of the separation of genetically bound electron-hole pairs. The activation energy of this process is found to be E _G =0.07 eV.     

Vibrational relaxation in liquids

A new theory is presented for vibrational energy relaxation in a liquid. It is shown that a vibrationally excited probe molecule relaxes through interaction with the density fluctuations in the surrounding solvent fluid. This interaction occurs through a potential V(k), which is expressed in terms of the intermolecular force between the excited probe molecule and the surrounding fluid molecules. By assuming spherically symmetric solvent particles the T ₁ energy relaxation time for direct V-T processes is related to the translational dynamic structure factor for the fluid S(k, ω_v), evaluated at the vibrational resonance frequency. It is shown that this is described by gas-like particle motions on a very short distance scale corresponding to k vectors lying well beyond the first or second peaks of the fluid structure factor S(k). Such motions can be pictured as high-frequency, short-distance distortions of the local equilibrium configuration of the solvent particles around the probe. T ₁ ^-1 is found to be proportional to ρ_e T ^1/2 ω_v ^-3. The V-V energy exchange relaxation time is also calculated. This is found to be proportional to S(k, ω′) evaluated at a frequency ω′, corresponding to the vibrational energy missmatch. An energy gap law for the V-V process is derived.     

Electron-electron interaction and instabilities in one-dimensional metals

The possible instabilities of a 1-dimensional itinerant electron gas are discussed, assuming electron-electron interaction to play the dominant role. As is well known, in the RPA, a 1-dimensional metal is prone to spin density wave (SDW), charge density wave (CDW) and Cooper pair (CP) instabilities. The spin channel decomposition of the irreducible scattering amplitude I is made and the spin channel projections are evaluated in terms of the matrix elements of bare electron-electron interactionV(x) for momenta of interest. It is found that if the bare electron interactionV(x) is repulsive and decreases monotonically with separation, only the SDW instability will occur. If the small separation (x?(2k _F )^?1) part of the interaction is greatly reduced or is made attractive,V(x) is non-monotonic,V _q (q?2k _F ) is negative, and a CDW instability is preferred. A CP instability is possible if the electron interaction is attractive,i.e., if [V _q (0<q<k _F )+V _q (q?2k _F )]<0. The above RPA results serve only as rough indicators, since in general there are important two-electron configurations with two-electron momentum close to zero and with electron hole momentum close to 2k _F , an example being the near Fermi energy configurationk ₁?k _F ,k ₂??k _F ,k ₃??k _F k ₄?k _F . Therefore as pointed out first by Bychkov, Gorkov and Dzhyaloshinskii (BGD), cross channel coupling is especially significant. It is shown that the cross channel coupling is constructive is some cases,eg., exchange of CD fluctuations leads to an effective electron-electron spin singlet attraction and vice-versa. A formalism for studying such effects is set up, and the particular example mentioned above is discussed. An RPA-like approximation is made for the form of the reducible singlet electron hole scattering amplitudeγ _s ^d and the resulting induced Cooper pair attraction is calculated to be $$\begin{gathered} [I_s ^e ]_{ind.} \rho _{{}^\varepsilon F} = [ln(\lambda \beta \omega _c )]^{ - 1} ln\{ [1 + 2\pi ^{ - 1} ln(\lambda \beta \omega _c )^2 ]/ \hfill \\ 1 + [8\pi ^{ - 1} \gamma _s ^d (q = 2k_F )^{ - 1} )^2 ]\} \hfill \\ \end{gathered} $$ where λ=1.14,β=(k _B T)^?1 andω ₀ is an electronic energy cut-off ～ε _F . The induced electron hole attraction due to the exchange of virtual Cooper pairs has a similar expression, but with a factor of (1/4) and withγ _s ^e (q=0) replacingγ _s ^d (q=2k _F ). The induced Cooper pair attraction is seen to be quite large over a broad range of temperatures close to but aboveT _CDW [i.e., aboveT such thatγ _s ^d (q=2k _F )^?1=0]. There is no requirement thatγ _s ^d (q=2k _F ) andγ _s ^e (q=0) become singular at the same temperature, as found by BGD. The BGD prediction is seen to arise from the neglect of real particle hole and particle-particle excitations while calculatingγ _s ^d andγ _s ^e . The effect of impurities, of electron-phonon coupling, of interchain coupling and of interaction between thermal order parameter fluctuations is discussed. The results are then applied to a discussion of the properties of TTF-TCNQ, where it is suggested that a CDW instability occurs becauseV _q (q=2k _F )<0,i.e., because the small separation electron repulsion is strongly reduced by the highly polarizable TTF. Because of substantial interchain coupling, the bulk CDW instability occurs close to the RPA instability temperature. The giant conductivity observed by Colemanet al is attributed to superconductive fluctuations in a 1-dimensional system with large mean field superconductive transition temperatureT _CP ^MF of order 300°K. Such a largeT _CP ^MF is shown to result from the induced Cooper pair attraction due to CD fluctuation exchange.     

Large deviations and Lifshitz singularity of the integrated density of states of random Hamiltonians

We consider the integrated density of states (IDS) ρ_∞(λ) of random Hamiltonian H_ω=?Δ+V_ω, V_ω being a random field on ? ^d which satisfies a mixing condition. We prove that the probability of large fluctuations of the finite volume IDS |Λ|^?1ρ(λ, H_Λ(ω)), Λ ? ? ^d , around the thermodynamic limit ρ_∞(λ) is bounded from above by exp {?k|Λ|},k>0. In this case ρ_∞(λ) can be recovered from a variational principle. Furthermore we show the existence of a Lifshitztype of singularity of ρ_∞(λ) as λ → 0⁺ in the case where V_ω is non-negative. More precisely we prove the following bound: ρ_∞(λ)≦exp(?kλ^?d/2) as λ → 0⁺ k>0. This last result is then discussed in some examples.     

Partial virial theorems for atomic-molecular systems

New virial relations for three-and four-particle atomic-molecular systems are proposed. Using operators of extension or squeezing of interparticle distances, it is shown that, for all pairs of j and k particles in S states of these systems, the following partial virial relations are valid: 〈2T _jk 〉+〈 V _jk 〉=0, where 〈V _jk 〉 is the average Coulomb interaction energy for a pair of particles and 〈T _jk 〉 is a part of the average kinetic energy of the system. There are three and six such relations for three-and four-particle systems, respectively. The conventional virial theorem (〈 2T〉+〈V〉=0) for the average total kinetic and potential energies of the system (〈 T〉 and 〈V〉, respectively) corresponds to the summation of partial virial relations over all pairs of particles. It is shown by an example of variational calculations of the helium atom ⁴He²⁺ e ^? e ^? and the helium muon-electron mesoatom ⁴He²⁺μ^? e ^? that partial virial relations are a highly sensitive indicator of the accuracy of wave functions.     

Optical and theoretical investigations of V ions in CdZnTe crystal

High-resistivity CdZnTe:V crystals are investigated by photoluminescence (PL) and by time-resolved PL in the infrared spectral range. A double peaked emission band is detected around 0.8 eV and it is related to vanadium doping. No-phonon lines of the internal transitions were detected. This emission is interpreted as a balance between the ⁴T₁(⁴P)→⁴T₁(⁴F) internal transition and an electronic transition from the conduction band to the ⁴T₁(⁴F) ground state of V²⁺. The corresponding decay time after laser excitation gives evidence to the contribution of two different recombination processes. These two emission bands are separated by time-resolved luminescence. Crystal-field calculations of the detected transition energies based on Tanabe-Sugano scheme are presented and the Racah parameter B and crystal-field intensity D_q were determined.In addition, a model is developed in terms of one-electron orbital, to explain the characteristics of the PL excitation processes of V²⁺. Excitations with above and below band edge energy confirm the proposed schemes.     

Spin-lattice relaxation of the Vk-center in LiF

The angular and temperature dependences of the spin-lattice relaxation (SLR) rate of V_k-centers in LiF doped with Mg or Ag have been investigated. In the temperature interval 4.2–100 K the results can be fitted by the formula $\text{τ}{}^{\mathrm{?1}}\text{= A(θ)T + Be}{}^{\mathrm{<mtext>?\Delta </mtext><mtext>T</mtext>}}$ with A(0°) = 0.11 sec^?1K^?1, A(90°) = 1.3 sec^?1K^?1, B = 3 × 10⁵sec^?1 and Δ = (175 ± 15)K.A mechanism for the SLR is considered, assuming the modulation of the hyperfine interaction by phononinduced transitions between the ground and excited states of the resonant molecular vibrations of the V_k-center. This mechanism is found to explain the value, the temperature dependence and the isotropy of τ^?1 in the interval T = 20–100 K.The one-phonon SLR mechanisms of the V_k-center in the T < 10 K region are discussed.     

NQR application to the study of hydrogen dynamics in hydrogen-bonded molecular dimers

The temperature dependences of ¹H NMR as well as ³⁵Cl NQR spin-lattice relaxation times T ₁ were investigated in order to study the hydrogen transfer dynamics in carboxylic acid dimers in 3,5-dichloro- and 2,6-dichlorobenzoic acids. The asymmetry energy A/ k _B and the activation energy V/ k _B for the hydrogen transfer were estimated to be 240 K and 900 K, and 840 K and 2500 K, respectively, for these compounds. In spite of a large asymmetric potential the quantum nature of hydrogen transfer is recognized in the slope of the temperature dependence of T ₁ on the low-temperature side of the T ₁ minimum. The NQR T ₁ measurements was revealed to be a good probe for the hydrogen transfer dynamics.     

A multi-channel scattering theory for some time dependent Hamiltonians,charge transfer problem

Scattering theory for time dependent HamiltonianH(t)=?(1/2) Δ+ΣV _j (x?q _j (t)) is discussed. The existence, asymptotic orthogonality and the asymptotic completeness of the multi-channel wave operators are obtained under the conditions that the potentials are short range: |V _j (x)|≦C _j (1+|x|)^?2?ε, roughly spoken; and the trajectoriesq _j (t) are straight lines at remote past and far future, and |q _j (t)?q _k (t)| → ∞ ast → ± ∞ (j≠k).     

Conversion coefficients and E0 transitions in102, 104, 106Pd

In in-beam (p, p′) experiments, electron and γ-spectra were measured in the electron energy range of 500-1840 keV for¹⁰²Pd and¹⁰⁴Pd, and 600–1580 keV for¹⁰⁶Pd. The conversion coefficients of all transitions in this range were obtained with accuracies of about 20%, in some favourable cases 10%. Special attention was given to 0⁺′-0⁺ transitions from the two-phonon triplets to the ground states with the following results for the branching ratios 0⁺′-0⁺ (ground state) to 0⁺′-2⁺ (one-phonon state):¹⁰²Pd:T _k (E0)/T _γ (E2)<(2.1±3.6)·10^{?7 104}Pd:T _k (E0)/T _γ (E2)=(6.0±1.4)·10^{?5 106}Pd:T _k (E0)/T _γ (E2)=(6.0±2.0)·10^?4     

Calculation of the wave functions of the ground and weakly excited states of helium II

The wave functions of the ground (Ψ₀) and the first excited (Ψ_k) states of He II in the second-order approximation, i.e., up to the first two corrections to the corresponding solutions for a weakly nonideal Bose gas, are determined by the collective variable method, which was proposed by Bogolyubov and Zubarev and developed in the studies by Yukhnovskii and Vakarchuk. The functions Ψ₀ and Ψ_k = ψ_kΨ₀ are determined as the eigenfunctions of the N-particle Schrödinger equation from a system of coupled equations for Ψ₀, Ψ_k, and the quasiparticle spectrum E(k) of helium II. The results consist in the following: (1) these equations are solved numerically for a higher order approximation compared with those investigated earlier (the first-order approximation), and (2) Ψ₀ and ψ_k are derived from a model potential of interaction between He⁴ atoms (rather than from the structure factor as earlier) in which the potential barrier is joined with the attractive potential found from experiment. The height V ₀ of the potential barrier is a free parameter. Except for V ₀, the model does not have any free parameters or functions. The calculated values of the structure factor, the ground-state energy E ₀, and the quasiparticle spectrum E(k) of He II are in agreement with the experimental values for V ₀ ≈ 100 K. The second-order correction to the logarithm of Ψ₀ significantly affects the value of E ₀ and provides the asymptotics E(k → 0) = ck, while the second-order correction to ψ_k slightly affects the E(k). The second-order corrections to Ψ₀ and ψ_k have a smaller effect on the results compared with the first-order corrections, whereby the theory is in agreement with experiment; therefore, one may assume that the truncated Ψ₀ and ψ_k well describe the microstructure of He II. Thus, the series for Ψ₀ and Ψ_k can be truncated in spite of the fact that the expansion parameter is not very small (～1/2).     

Lift order problems for ordinary differential equations on manifolds

Fork≥0, let ττ_k:T ^k+1(M)=T(T ^k(M))→T ^k(M) denote the (k+1)th iterated tangent bundle in relation to a base manifoldT ⁰(M)=M. LetV represent a possibly nonstationary vector field overT ^k(M), and letQ be a subset/submanifold inT ^k(M). Sufficient conditions (and, whenV is completely integrable inQ, necessary and sufficient conditions) are established to ensure that all solutionsg toy′=V(t, y) lying entirely inQ have the formG=f ^[k], wheref ^[k] is thekth-order differential lift of a curvef lying inM. The relevance of the issue for higher order dynamical systems (especially in mechanics) is discussed. Higher order involutions and complete vector field lifts are examined from the viewpoint of the differential equations they present. Collateral results on the general solvability of initial value problems are obtained and numerous examples are discussed in detail. To the memory of my teacher and friend M. Kuga (1928–1990).     

Electron density of states and superconducting Tc in A15-compounds

We compare the superconductivity parameters of V₃Ga and V₃Ge by the method of McMillan with the help of new neutron diffraction data obtained by P. Schweiss. We consider the strongly varying density of states in V₃Ga at the Fermi energy, a complication suggested by Labbé, Barisic, and Friedel. We find that increased electronic density of states contributes about equally with lattice softening to the enhancement of T_c in V₃Ga as compared to V₃Ge, although the values of N(O)J² = Mλ〈ω²〉 are roughly consistent with those obtained for some other metals by McMillan. The reason N(O) is not more effective in raising T_c is the partial compensation obtained from averaging over the narrow density-of-states peak which gives N ? 0.6N(O).